Definition:Orthonormal Subset

Definition

Let $\struct {V, \innerprod \cdot \cdot}$ be an inner product space.

Let $S \subseteq V$ be a subset of $V$.

Then $S$ is an orthonormal subset if and only if:

$(1): \quad \forall u \in S: \norm u = 1$

where $\norm {\, \cdot \,}$ is the inner product norm.

$(2): \quad S$ is an orthogonal set:

$\forall u, v \in S: u \ne v \implies \innerprod u v = 0$

Also see

• Definition:Basis (Hilbert Space)
• Orthonormal Subset of Hilbert Space Extends to Basis

Sources

• 1990: John B. Conway: A Course in Functional Analysis ... (previous) ... (next) $\text I.4.1$
• 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 2$: Riemannian Metrics. Definitions